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        <h1 id="数学"><a href="#数学" class="headerlink" title="数学"></a>数学</h1><h2 id="1"><a href="#1" class="headerlink" title="1"></a>1</h2><h3 id="极限"><a href="#极限" class="headerlink" title="极限"></a>极限</h3><ul>
<li><p>核心考点</p>
<ul>
<li>定义</li>
<li>性质</li>
<li>计算</li>
<li>应用</li>
</ul>
</li>
<li><p>定义</p>
<ul>
<li></li>
</ul>
</li>
</ul>
<h2 id="2"><a href="#2" class="headerlink" title="2"></a>2</h2><h2 id="3-一元函数微分学"><a href="#3-一元函数微分学" class="headerlink" title="3 一元函数微分学"></a>3 一元函数微分学</h2><h3 id="一、导数定义"><a href="#一、导数定义" class="headerlink" title="一、导数定义"></a>一、导数定义</h3><ul>
<li><p>\text{1.定义} \lim_{\bigtriangleup x \to 0} \frac{f(x_0+\bigtriangleup x)-f(x_0)}{\bigtriangleup x}=f’(x_0)\text{   瞬时变化率} \\<br>\text{[注]①换元法：令}x_0+\bigtriangleup x=x\Rightarrow  \lim_{\bigtriangleup x \to 0} \frac{f(x)-f(x_0)}{x-x_0} \\<br>\text{②左右导数}\begin{cases}<br>\lim_{\bigtriangleup x \to 0^+} \frac{f(x_0+\bigtriangleup x)-f(x_0)}{\bigtriangleup x}=f’_+(x_0) \\<br>\lim_{\bigtriangleup x \to 0^-} \frac{f(x_0+\bigtriangleup x)-f(x_0)}{\bigtriangleup x}=f’_-(x_0)<br>\end{cases} \\<br>\therefore f(x_0)\exists \Leftrightarrow f’(x_0)=f’_+(x_0)=f’_-(x_0)</p>
</li>
<li><script type="math/tex; mode=display">
\begin{cases}
\text{定义法} \\
\text{公式法}
\end{cases} \\
\text{1）抽象f(x)在一点}
\begin{cases}
\text{泛指x_0} \\
\text{暗指x_0}
\end{cases} \\
\text{2）分段函数在分段点(导数定义)} \\
\text{3）四则运算} 
\begin{cases}
\text{太复杂的点}
\begin{cases}
\text{f=f_1+f_2} \\
\text{f=f_1f_2f_3}
\end{cases} \\
\text{不成立的点}
\end{cases}</script></li>
</ul>
<h4 id="例子"><a href="#例子" class="headerlink" title="例子"></a>例子</h4><p><img src="1 (2" alt="1 (2)">.jpg)<br><img src="IMG_20210821_132841.jpg" alt="IMG_20210821_132841"><br><img src="1 (2" alt="1 (2)">-1642941232230.jpg)</p>
<h3 id="二、计算与几何应用"><a href="#二、计算与几何应用" class="headerlink" title="二、计算与几何应用"></a>二、计算与几何应用</h3><h4 id="1-基本公式"><a href="#1-基本公式" class="headerlink" title="1.基本公式"></a>1.基本公式</h4><script type="math/tex; mode=display">
(x^k)'=kx^{k-1} \\
(ln x)'=\frac{1}{x}  \\
(e^x)'=e^x \\
(\sin x)'=\cos x \\
(\cos x)'=-\sin x \\
(\tan x)'= \sec^2 x\\
(\cot x)'= -\csc^2 x\\
(\sec x)'= \sec x\tan x\\
(\csc x)'= -\csc x\cot x\\
(\arctan x)'= \frac{1}{1+x^2} \\
(arccot x)'= -\frac{1}{1+x^2}\\
(\arcsin x)'= \frac{1}{\sqrt{1-x^2} }\\
(\arccos x)'= -\frac{1}{\sqrt{1-x^2} } \\
(\ln{x+\sqrt{x^2+a^2}})'= \frac{1}{\sqrt{x^2+a^2}}\\
(\ln{x+\sqrt{x^2-a^2}})'= \frac{1}{\sqrt{x^2-a^2}}\\
\text{变限中}(F(x)=\int_{\upsilon _2(x)}^{\upsilon _1(x)}f(t)dt  \\
F'(x)=f[\upsilon _2(x)]\upsilon _2'(x)-f[\upsilon _1(x)] \upsilon _1'(x)\\</script><h4 id="2-1"><a href="#2-1" class="headerlink" title="2."></a>2.</h4><h2 id="4"><a href="#4" class="headerlink" title="4"></a>4</h2><h2 id="6"><a href="#6" class="headerlink" title="6"></a>6</h2><h3 id="微分方程"><a href="#微分方程" class="headerlink" title="微分方程"></a>微分方程</h3><ul>
<li><p>计算</p>
<ul>
<li><p>一阶微分方程的求解</p>
<ul>
<li></li>
<li><p>子主题 2</p>
<ul>
<li>子主题 3</li>
</ul>
</li>
<li><p>子主题 5</p>
</li>
<li>子主题 6</li>
<li>子主题 5</li>
</ul>
</li>
<li><p>二阶可降解微分方程</p>
</li>
<li>高阶常系数线性微分方程</li>
<li><p>换元法求解微分方程</p>
<ul>
<li>求导公式逆用换元</li>
<li>自变量换元</li>
<li>因变量</li>
<li>xy地位互换换元</li>
</ul>
</li>
</ul>
</li>
<li><p>应用</p>
<ul>
<li>用极限导数定义建方程</li>
<li><p>用几何应用建方程</p>
<ul>
<li>曲线切线斜率</li>
<li>f1f2的公切线</li>
<li>截距</li>
<li>面积体积平均值弧长侧面积曲率质心</li>
</ul>
</li>
<li><p>用变化建方程</p>
<ul>
<li>元素衰变问题</li>
<li>人口增长</li>
<li>物线追踪</li>
<li>冷却问题</li>
<li>牛二</li>
</ul>
</li>
</ul>
</li>
</ul>
<h2 id="10-一元函数积分——-几何应用"><a href="#10-一元函数积分——-几何应用" class="headerlink" title="10 一元函数积分—— 几何应用"></a>10 一元函数积分—— 几何应用</h2><h3 id="知识图谱"><a href="#知识图谱" class="headerlink" title="知识图谱"></a>知识图谱</h3><script type="math/tex; mode=display">
\begin{cases}
\text{研究对象} \begin{cases}
 f(x)\\
 fn(x)\begin{cases}
x=x(t) \\
y=y(t)
\end{cases}\\
\frac{\partial f}{\partial x}  \\
\int_{a}^{x}f(t)dt  \\
\text{微分方程的解f(x)}
\end{cases}\\
\text{研究内容}\begin{cases}
\text{面积：计算\int_{a}^{b}f(x)dx   \\
\text{旋转体体积} \\
\text{平均值} \int _{a}^{b}{\frac{f(x)dx}{b-a}}   \\
\text{平面曲线的弧长} \\
\text{旋转曲面的面积} \\
\text{平面上的曲边梯形} \\
\text{平行截面面积为已知的立体图形} 
\end{cases}
\end{cases}</script><h3 id="研究对象"><a href="#研究对象" class="headerlink" title="研究对象"></a>研究对象</h3><h4 id="f-x"><a href="#f-x" class="headerlink" title="f(x)"></a>f(x)</h4><h4 id="fn-x"><a href="#fn-x" class="headerlink" title="fn(x)"></a>fn(x)</h4><h5 id="x-x-t"><a href="#x-x-t" class="headerlink" title="x=x(t)"></a>x=x(t)</h5><h5 id="y-y-t"><a href="#y-y-t" class="headerlink" title="y=y(t)"></a>y=y(t)</h5><h4 id="微分方程的解f-x"><a href="#微分方程的解f-x" class="headerlink" title="微分方程的解f(x)"></a>微分方程的解f(x)</h4><h3 id="研究内容"><a href="#研究内容" class="headerlink" title="研究内容"></a>研究内容</h3><h4 id="面积"><a href="#面积" class="headerlink" title="面积"></a>面积</h4><h4 id="旋转体体积"><a href="#旋转体体积" class="headerlink" title="旋转体体积"></a>旋转体体积</h4><h4 id="平均值"><a href="#平均值" class="headerlink" title="平均值"></a>平均值</h4><h4 id="平面曲线的弧长"><a href="#平面曲线的弧长" class="headerlink" title="平面曲线的弧长"></a>平面曲线的弧长</h4><h4 id="旋转曲面的面积"><a href="#旋转曲面的面积" class="headerlink" title="旋转曲面的面积"></a>旋转曲面的面积</h4><h4 id="平面上的曲边梯形"><a href="#平面上的曲边梯形" class="headerlink" title="平面上的曲边梯形"></a>平面上的曲边梯形</h4><h4 id="平行截面面积为已知的立体图形"><a href="#平行截面面积为已知的立体图形" class="headerlink" title="平行截面面积为已知的立体图形"></a>平行截面面积为已知的立体图形</h4><h2 id="15"><a href="#15" class="headerlink" title="15"></a>15</h2><h3 id="微分方程-1"><a href="#微分方程-1" class="headerlink" title="微分方程"></a>微分方程</h3><ul>
<li>一阶微分方程的求解</li>
</ul>
<h2 id="16"><a href="#16" class="headerlink" title="16"></a>16</h2><h3 id="无穷级数"><a href="#无穷级数" class="headerlink" title="无穷级数"></a>无穷级数</h3><ul>
<li><p>数列级数的判敛</p>
<ul>
<li>定义 Sn</li>
<li><p>判敛法</p>
<ul>
<li>正向级数</li>
<li>交错级数</li>
<li>任意项级数</li>
</ul>
</li>
<li><p>常用结论</p>
</li>
</ul>
</li>
<li><p>幂级数的收敛域</p>
<ul>
<li><p>概念</p>
<ul>
<li>幂级数</li>
<li>收敛点与发散点</li>
</ul>
</li>
<li><p>具体性问题</p>
<ul>
<li>anxn</li>
<li>缺项性问题</li>
</ul>
</li>
<li><p>抽象性问题</p>
<ul>
<li>阿贝尔定理</li>
<li>结论1</li>
<li>结论2</li>
</ul>
</li>
</ul>
</li>
<li><p>展开问题</p>
<ul>
<li><p>考法</p>
<ul>
<li>函数展开</li>
<li>积分展开</li>
<li>导数展开</li>
<li>无穷小比阶</li>
</ul>
</li>
<li><p>攻工具</p>
<ul>
<li>先积后导</li>
<li>先导厚积</li>
<li>重要展开公式</li>
</ul>
</li>
</ul>
</li>
<li><p>求和问题</p>
<ul>
<li>直接套公式</li>
<li>用先积后导或先导厚积求和函数</li>
<li>用所给微分方程求和函数</li>
<li>建立微分方程并求和函数</li>
<li>综合题</li>
</ul>
</li>
<li><p>傅里叶级数</p>
<ul>
<li>迪利克雷收敛</li>
<li>周期为2I的周期函数的傅里叶级数与系数公式</li>
</ul>
</li>
</ul>
<h2 id="分支主题-15"><a href="#分支主题-15" class="headerlink" title="分支主题 15"></a>分支主题 15</h2><h2 id="分支主题-13"><a href="#分支主题-13" class="headerlink" title="分支主题 13"></a>分支主题 13</h2><h2 id="分支主题-14"><a href="#分支主题-14" class="headerlink" title="分支主题 14"></a>分支主题 14</h2>
      
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